All of the 3rd grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$49.50$ in total. The next month, the same group visited a science museum where the tickets cost $$25.50$ each for teachers and $$8.50$ each for students, and the group paid $$144.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 49.5}$ ${25.5x+8.5y = 144.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-9y = -148.5}$ ${25.5x+8.5y = 144.5}$ Add the top and bottom equations together. $ -0.5y = -4 $ $ y = \dfrac{-4}{-0.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+3y = 49.5}$ to find $x$ ${8.5x + 3}{(8)}{= 49.5}$ $8.5x+24 = 49.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {25.5x+8.5y = 144.5}$ and get the same answer for $x$ ${25.5x + 8.5}{(8)}{= 144.5}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.